Apple's "Bedtime" isn't a sleep tracking app, really. It's more of a "convenient" way to set bedtime and waking alarms, so it's not a great comparison.
In answer to the OP, I actually used to use a Zeo headband EEG that I loved. Now I just use my Apple Watch, along with an app called AutoSleep. If you don't wear your watch to bed, it just keeps track of the time you aren't using your phone or watch, and assumes that's sleep time, which it records historically. If you do wear your watch to bed, it tracks your heart rate and movement as a proxy for sleep quality. The nice thing is that it doesn't require user intervention to start or stop, but it does require you throw your watch on a charger for fifteen minutes or a half an hour in the morning, and again in the evening, to ensure your watch is perpetually charged to make it through the day and the night.
